Separation of a diagonal (Q1044866)

Given a topological space \(X\), the diagonal \(\Delta_X\) of \(X\) is defined by \(\Delta_X = \{\langle x, x \rangle \in X \times X: x\in X \}\). In this paper, the author systematically investigates several separation properties defined in terms of the diagonal of a space, namely, \(\Delta\)-normality, functional \(\Delta\)-normality, \(\Delta\)-paracompactness, and regular \(\Delta\)-paracompactness. Some relationships among these and other topological properties such as normality and paracompactness are established. For example, it is shown that paracompact spaces are functionally \(\Delta\)-paracompact, and functionally \(\Delta\)-paracompact spaces are functionally \(\Delta\)-normal. It is also shown that GO-spaces are functionally \(\Delta\)-paracompact, and regular \(\Delta\)-paracompact spaces are normal. Although it is known that the space \((\omega_1+1)\times (\omega_1+1) \setminus \{\omega_1, \omega_1\}\) is \(\Delta\)-paracompact but not normal, it is still unclear whether every functionally \(\Delta\)-paracompact, or every \(\Delta\)-paracompact, or every \(\Delta\)-normal space must be normal. Several partial results to these questions are given. In addition to these, the author also poses many other open questions on this topic.